For a US airline company you do. Thought we were discussing pilot jobs not gas station employees or bank workers who work for the UK branch of Merrill Lynch or whomever. Obviously, baggage handlers, ticket sales, check-in workers are locally employed via a local contract handled through nation-to-nation labor rules.
Unless the airline forms an international branch which, to my knowledge, none have. Happy to hear of any examples of hiring local (non-US or non Green Card holder) pilots and basing them at the destination. Possibly Fedex with the Hkg base but all the guys I know who fly for Fedex out of Hkg are yanks.
Plus, doubt if the US pilot unions would allow it regardless.